statistique, théorie et gestion de portefeuille - Docs at ISFA

This yields
×
h+
Sc−2
− 2 dh e ik
Sc−2 M(2)−1 w wt
M (2)h+
ik
Sc−2 M(2)−1
S c ( wiσi ) c−1
P (S) ∝ e −
⎡
⎣1 + Sc−3
M
6
(3)
dk k2
e− 2S
2π c−2 wtM (2)−1w ×
ijk
ijk hihjhk + · · ·
⎤
415
⎦ . (167)
Denoting by 〈·〉h the average with respect to the Gaussian distribution of h and by 〈·〉k the average with
respect to the Gaussian distribution of k, we have :
×
⎡
⎣1 + Sc−3
6
ijk
P (S) ∝
M (3)
ijk 〈〈hihjhk〉h〉k + Sc−4
24
det M (2)−1
w t M (2)−1 w (2πS2−c ) N−1
2 e −
S c ( wiσi ) c−1 ×
M (4)
ijkl 〈〈hihjhkhl〉h〉k
⎤
+ · · · ⎦ . (168)
We now invoke Wick’s theorem 2 , which states that each term 〈〈hi · · · hp〉h〉k can be expressed as a product
of pairwise correlation coefficients. Evaluating the average with respect to the symmetric distribution of k,
it is obvious that odd-order terms will vanish and that the count of powers of S involved in each even-order
term shows that all are sub-dominant. So, up to the leading order :
P (S) ∝
The matrix M (2) can be calculated, which yields
M (2)
kl
and shows that
=
=
ijkl
det M (2)−1
w t M (2)−1 w (2πS2−c ) N−1
2 e −
S c ( wiσi ) c−1 . (169)
1
( wiσi) c−2
c
wk
c − 1 δkl +
2 σk
c2 c
−1 2 V
2 kl σ −1
k σ c
2 −1
l , (170)
1
( wiσi) c−2 ˜ Mkl, (171)
det M (2)−1
w t M (2)−1 w =
c
(N−1)( 2
wiσi
−1)
The inverse matrix ˜ M −1 satisfies
l ˜ Mkl · ( ˜ M −1 )lj = δkj which can be rewritten:
or equivalently
c
wk
c − 1 (
2 σk
˜ M −1 )kj + c2
2
c
c − 1 wk (
2 ˜ M −1 )kj + c2
2
V −1
kl · ( ˜ M −1 )ljσ c
k
l
V −1
kl · ( ˜ M −1 )ljσ c
2
k
l
det ˜M −1
wt ˜M −1 . (172)
w
2 −1
σ c
2 −1
l = δkj (173)
σ c
2 −1
l = δkj · σk (174)
2 See for instance (Brézin et al. 1976) for a general introduction, (Sornette 1998) for an early application to the portfolio problem
and (Sornette et al. 2000b) for a systematic utilization with the help of diagrams.
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